In this paper, we view the above as merely the beginning of the story and aim to additionally construct moduli spaces of unstable objects. Instability in reductive GIT is a structured phenomenon: for each unstable point of \(Y\) , there is a maximally destabilising 1-PS of \(G\) , unique up to conjugation under a parabolic subgroup, which is most responsibleThis notion depends on a choice of norm on 1-PSs of \(G\) . for violating the Hilbert–Mumford criterion [1]}; this is akin to the Harder–Narasimhan filtration [2]} of a vector bundle being a maximally destabilising filtration. These maximally destabilising 1-PSs gives rise to an instability stratification (or HKKN stratification following [3]}, [1]}, [5]}, [6]}) of \(Y\)  into finitely many \(G\) -invariant locally closed subschemes such that the lowest stratum \(S_0\)  is the semistable set and the higher (unstable) strata \(S_\beta \)  are indexed by conjugacy classes \(\beta \)  of rational 1-PSs of \(G\)  which are maximally destabilising for the points in \(S_\beta \) . Recently, Halpern-Leistner greatly generalised this idea to \(\Theta \) -stratifications of stacks [7]}. The HKKN stratification suggests that the instability types \(\beta \)  should be viewed as discrete invariants and one should then look to construct moduli spaces for unstable objects of instability type \(\beta \) . For the GIT construction of moduli of vector bundles on a curve, GIT instability types correspond to Harder–Narasimhan (HN) types [8]}, [9]}.
